On Supercompactly and Compactly Generated Toposes
A final draft of my article on supercompactly generated toposes has gone up on the arXiv. Due to it's length, I intend to submit it to the new Expositions journal in TAC; in the mean time, any comments are most welcome. Here is a summary:
A supercompact object in a topos is one which is simple or irreducible in the sense that any covering of it (any jointly epimorphic family of morphisms with this object as their codomain) must contain an epimorphism. A compact object is defined similarly, except that every covering is instead required to contain a finite subcover. These concepts arise very naturally in the setting of my ongoing work with toposes of monoid actions, because these objects correspond to the principal (aka cyclic) and finitely generated monoid actions respectively.
While proving results in my ongoing work about properties of toposes of topological monoid actions, I realised that many of the results relied only on the fact that these toposes have enough supercompact objects: every monoid action is a union of its principal sub-actions. As the volume of results exceeded what I strictly needed for studying monoid actions, I decided to devote a paper to toposes with this property, which turn out to be more common than you might expect. It turns out that toposes with enough compact objects behave similarly enough that I was able to explore them in parallel. Writing this paper also gave me the opportunity to present explicitly some special cases of the results in Prof Caramello's paper from summer of last year, which has lots of details about the relationships between sites and (co)morphisms of sites and the corresponding toposes and geometric morphisms.
I found some pleasing duality results and a bunch of examples which I hope people will find interesting.